Large Deviations for Markov Chains

This book explores the large deviations for empirical measures and vector-valued additive functionals of Markov chains, providing asymptotic estimates of probabilities of deviating from ergodic behavior.

Format: Hardback
Length: 230 pages
Publication date: 27 October 2022
Publisher: Cambridge University Press

This comprehensive book delves into the realm of analyzing large deviations for empirical measures and vector-valued additive functionals of Markov chains with general state spaces. By imposing suitable recurrence conditions, the ergodic theorem for additive functionals of a Markov chain establishes the almost sure convergence of the averages of a real or vector-valued function of the chain to the mean of the function with respect to the invariant distribution. In the case of empirical measures, the ergodic theorem guarantees the almost sure convergence in a suitable sense to the invariant distribution. Furthermore, the large deviation theorems offer precise asymptotic estimates at the logarithmic level of the probabilities of deviating from the preponderant behavior predicted by the ergodic theorems.

Introduction:
Markov chains are a fundamental tool in probability theory and statistical mechanics, allowing us to model and analyze complex systems. They are characterized by a transition matrix that governs the probability of moving from one state to another over time. The study of large deviations for empirical measures and vector-valued additive functionals of Markov chains has gained significant attention in recent years due to its importance in understanding the behavior of complex systems.

Ergodic Theorem for Additive Functionals:
The ergodic theorem for additive functionals of a Markov chain asserts the almost sure convergence of the averages of a real or vector-valued function of the chain to the mean of the function with respect to the invariant distribution. In other words, as the time horizon increases, the average of the function of the Markov chain tends to approach the expected value of the function, assuming the initial distribution is invariant. This theorem is a powerful tool for analyzing the long-term behavior of Markov chains and has numerous applications in fields such as finance, biology, and social sciences.

Empirical Measures:
In the case of empirical measures, the ergodic theorem states the almost sure convergence in a suitable sense to the invariant distribution. This convergence is achieved by considering the empirical process that generates the empirical measures. The empirical process is a stochastic process that follows the Markov chain, and the empirical measures are the random variables that are generated by this process. The ergodic theorem guarantees that the empirical measures will converge to the invariant distribution as the time horizon increases, assuming the initial distribution is invariant.

Large Deviation Theorems:
The large deviation theorems provide precise asymptotic estimates at the logarithmic level of the probabilities of deviating from the preponderant behavior asserted by the ergodic theorems. These theorems are particularly useful in situations where the preponderant behavior is difficult to predict or where the system is subject to sudden changes. By analyzing the large deviations, we can gain insights into the rare events that occur in the system and develop strategies for mitigating their impact.

Applications:
The applications of large deviation theorems are diverse and wide-ranging. In finance, they are used to analyze the risk of financial portfolios and to develop risk management strategies. In biology, they are used to study the evolution of populations and to identify genetic mutations that contribute to disease. In social sciences, they are used to analyze social networks and to predict social phenomena such as outbreaks and revolutions.

Conclusion:
In conclusion, this book provides a comprehensive and in-depth exploration of the large deviations for empirical measures and vector-valued additive functionals of Markov chains. It covers the ergodic theorem for additive functionals, empirical measures, large deviation theorems, and their applications in various fields. By understanding the behavior of complex systems through the lens of large deviations, we can gain insights into the rare events that occur in the system and develop strategies for mitigating their impact. This book is a valuable resource for researchers, practitioners, and students interested in probability theory, statistical mechanics, and applied mathematics.

Weight: 544g
Dimension: 159 x 235 x 26 (mm)
ISBN-13: 9781316511893